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A340465 Primes of the form prime(i)*prime(i+1)+prime(i+2)*prime(i+3)+...+prime(k-1)*prime(k).

%I A340465 #15 Jan 10 2021 22:11:04
%S A340465 41,313,2137,6569,7853,10133,10847,12401,13757,14747,17569,17911,
%T A340465 24001,24049,27901,31307,38729,43177,43961,44819,51607,69191,81517,
%U A340465 88379,104683,107099,130631,137177,138239,145967,154487,154723,158777,162947,175463,184409,192853,196169,232499,243137,261983
%N A340465 Primes of the form prime(i)*prime(i+1)+prime(i+2)*prime(i+3)+...+prime(k-1)*prime(k).
%C A340465 A prime that has more than one expression of the given form is included only once.  The first such prime is a(14353) = 6858604873 = 1979*1987+...+7109*7121 = 19949*19961+...+20231*20233.
%H A340465 Robert Israel, <a href="/A340465/b340465.txt">Table of n, a(n) for n = 1..10000</a>
%e A340465 a(1) = 2*3+5*7 = 41.
%e A340465 a(2) = 3*5+7*11+13*17 = 313.
%e A340465 a(3) = 17*19+23*29+31*37 = 2137.
%e A340465 a(4) = 5*7+11*13+17*19+23*29+31*37+41*43+47*53 = 6569.
%e A340465 a(5) = 41*43+47*53+59*61 = 7853.
%p A340465 S1:= [0,seq(ithprime(2*i)*ithprime(2*i+1),i=1..100)]:
%p A340465 P1:= ListTools:-PartialSums(S1):
%p A340465 S2:= [0,seq(ithprime(2*i-1)*ithprime(2*i),i=1..100)]:
%p A340465 P2:= ListTools:-PartialSums(S2):
%p A340465 M:= 2*max(S1):
%p A340465 S:= select(t -> t < M and isprime(t), {seq(seq(P1[i]-P1[j],j=i mod 2 + 1 .. i-2,2),i=1..101)} union {seq(seq(P2[i]-P2[j],j=i mod 2 + 1..i-2,2),i=1..101)} union {seq(P2[i],i=1..101,2)}):
%p A340465 sort(convert(S,list));
%o A340465 (Python)
%o A340465 from sympy import isprime, nextprime, prime
%o A340465 def sp2(lst):
%o A340465   ans = 0
%o A340465   for i in range(0, len(lst), 2): ans += lst[i]*lst[i+1]
%o A340465   return ans
%o A340465 def aupto(nn):
%o A340465   alst, i = [], 1
%o A340465   while True:
%o A340465     consec2i = [prime(j+1) for j in range(2*i)]; sp = sp2(consec2i)
%o A340465     if sp > nn: break
%o A340465     while sp <= nn:
%o A340465       if isprime(sp): alst.append(sp)
%o A340465       consec2i = consec2i[1:] + [nextprime(consec2i[-1])]; sp = sp2(consec2i)
%o A340465     i += 1
%o A340465   return sorted(alst)
%o A340465 print(aupto(261983)) # _Michael S. Branicky_, Jan 08 2021
%Y A340465 Includes A340464.
%K A340465 nonn
%O A340465 1,1
%A A340465 _J. M. Bergot_ and _Robert Israel_, Jan 08 2021